Financial Intermediation at Any Scale
For Quantitative Modelling (1/3)
Cours BachelierCharles-Albert Lehalle
Capital Fund Management, Paris and Imperial College, London
IHP, November 18, 2016 to December 4, 2016

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Motivation

What about the topic “Intermediation on Financial Markets”

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Since the 2008-2009 crisis legislators’ and regulators’ viewpoint on financial markets changed,

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They target to monitor and limit the risk taken by the market participants,

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In one sentence: they want to ensure most participants plays a role of intermediaries , and nothing more.

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The notion of intermediation and the role of banks, investment banks, dealers, brokers, and now insurancecompanies and funds have evolved and continue to evolve;I

important concepts to understand this are: microstructure and infrastructure; they are linked toliquidity .

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These last 10 years, the field of Market Microstructure emerged. Related literature has grown...

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I am convinced financial mathematics can address quite efficiently core concepts, as partly an academic
and partly a professional, I dedicated the last 12 years to understand these changes from a practical and atheoretical viewpoint.

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These sessions will be the occasion to share how, in my opinion, financial mathematics can answer to new
and important questions raised by recent changes.

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The Standard Pitch(We Will Go Further Than This)

Following the 2008 crisis, the financial system changed a lot:I

“Clients” (from inside or outside) have no more appetite for sophisticated products.

⇒ The system went from a bespoke market to a mass market.
Bespoke means to sell products that are very different: no economies of scale but high margins.
Mass market means a lot of similar products + optimized logistics.
I

Regulators welcome this change because it can prevent an accumulation of risk in inventories (cf. optimized
logistics).

⇒ The G20 of Pittsburgh (Sept. 2009) put the emphasis on inventory control (it is the root of improved
clearing, segregated risk limits, etc).
⇒ Policy makers took profit of two existing regulations (Reg NMS in the US and MiFID in Europe) to pushtoward electronification of exchanges (i.e. improved traceability and less information asymmetry).
I

Technology went into the game. Think about the kind of recent “innovations” (uber, booking.com, M-pesa,
blockchain, etc): it is about disintermediation .

⇒ How do you desintermediate a system made of intermediates?
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Market Microstructure?

Historically, market microstructure stands for not reducing
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Sellers = Equity Shares and Bonds issuers

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Buyers = investors.

In practice, today, associated topics are
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Market impact, Fire sales and Flash Crashes

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Auction / Matching mechanisms (Limit Orderbooks, RFQ, conditional / fuzzy matching, etc)

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Optimal trading / Liquidation

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Market Making and High Frequency Trading

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Investment process while taking all this into account

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My Viewpoint on Market Microstructure
I have been Global Head of Quantitative Research at Crédit Agricole Cheuvreux and CIB during years (including
the crisis). I discuss a lot with regulators; previously inside the working group on Financial Innovation of the
ESMA, now inside the Scientific Committee of the AMF. I am now in a large Hedge Fund.
I

From a Financial
Mathematics
perspective, it isnothing more thanadding a variable toour models: theLiquidity .

I

The interactions
between liquidity andother (usual) variables
is far from trivial.

Disclaimer : I express myown opinion and not the oneof any of these institutions.
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What Will We Do During 3 Lectures + 1 Seminar?
I will not go in the details of the models (except for few of them), because I target to give you enough information
to include liquidity in the models you know better than me.
Hence, I will
+ 18 Nov:
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Start by the definition of intermediation

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Focus on the two main Liquidity variables on financial market: inventories and flows+ 25 Nov:
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Show you what Liquidity looks like when we can observe it+ 2 Dec:
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Underline why market making (inventory keeping) and optimal trading (flow management) are core for the
new role of market participants.

+ 2 Dec [Seminar]:
I

Explain what practitioners are doing.

It is an on-going work
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Advertisement For Optimal Trading

Optimal Trading is About To Close The loop

My own viewpoint on optimal trading:I

We have sophisticated (but tractable) methods to optimize the strategy of one agent (investment bank,trader, asset manager, etc) facing a “background noise” (stochastic control is now really mature),

I

These methods are used by practitioners (already three books on this topic [Lehalle et al., 2013],[Cartea et al., 2015], [Guéant, 2016]),I

Differential games, and more specifically mean field games now propose very promising frameworks toreplace most of the background noise by a mean field of explicitly modelled agents:II

to provide robust results for practitioners [Cardaliaguet and Lehalle, 2016],
to obtain meaningful results for policy recommandations [Lachapelle et al., 2016].

Up to now most results on global modelling used a simplification of a reality. Now decisions are modelled andsystematic, why not inject them into a global model?
It should enable you to produce very accurate models and draw powerful conclusions.
I

Beyond optimal trading, these lectures should help you in introducing liquidity in any model of yours: pleaseask question!

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Outline of the Sessions

1

The Financial System as a Network of Intermediaries

2

Stylized Facts on Liquidity

3

Optimal Trading

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Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

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Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

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Risk Transformation 101

To understand the interactions between actors of financial markets, a first step is to understand therole of the financial system .
It takes its role at the root of capitalism:I

say you see a shoes shiner at Deli, IndiaI

you pay $1 to have your shoes shined, and you ask to the guy

I“it seems you have around 30 customers each day, it let you with $30 every day, it is a good job.”I

he answers: “not at all, I earn $1 a day... I do not own the brush, its owner loans its to me $29 a day. Since a
brush costs $12 and I need my daily dollar to eat, I will never own one.”→ let’s discuss about microcredit: loan him $12 during 2 days...
You have $30, you can ask to the guy some percents to cover the risk he will not have enough clients. If you arerisk averse, you can even ask for the brush as collateral... A bank can “structures” the loan for you, it will takecare of all the administrative aspects, it is a simple risk transformation (liquidity on you side, business of theshoes shiner side).

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Intermediation 101

If you want to implement microcredit on your own, it is impossible: you will never cross the path of someone
in the situation of the shoes shiner. The bank can find borrowers and lenders. As an intermediary, it should do
two thinksI

concentrate the flow, building a marketplace ;I

provide neutral information to both sides of the loan;

Itake care of collateral ;it deserves fees that for.
The bank can even have an incentive to make the market : if a borrower is there but not lender is present, it
can provide the loan itself, and wait for the next lender.
⇒ how can you make the difference between a bank waiting for the next three lenders and a bank taking
directional risk itself?

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The Subtle Balance of Intermediaries’ Inventories
If we take the viewpoint of looking at the financial network from the outside , we need to understands its inputs
and outputs, and deduce the features it provides to the rest of the economy.
I

We can see how the banking network operates a maturity transformation between natural borrowers at
different maturities (mid or long term) and natural lenders (short or mid term).

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Investment banks operate the same way with a lot of other risk transformation (insurance –ie optional
payoffs–, structured products, swaps, etc),

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Banks are intermediaries: they have no reason to keep risk in their inventories.

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The bad cases are when all the banks host risks in the same direction (2008), instead of having adiversification at the scale of the whole system.

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Banks are often tempted to take directional risk (sometimes without really knowing it). It is the goal of
regulators to force them to maintain the risk of their inventory as low as possible (using capitalrequirements).

I

Nevertheless inflows and outflows in Banks balance sheets (i.e. transactions) are not simultaneous, hence
regulators need to give them some freedom to wait for a seller once they sold a contract to a buyer (and thereverse).

Let’s see this viewpoint is typically a microstructural one: intermediaries, buyers and sellers, inventory risk...
Most of the buzz words are there...
CA Lehalle (Cours Bachelier, 2016)10 / 63More Sophisticated: Risk Trading Inside the System

On the one side (right) you have Cash PMs they arecash rich but safety poor (“fear to loose their money”).
On the other side (left) you have Risk PMs , they haveto beat a benchmark, thus are securities rich but returnpoor (the need leverage and non linearities).
In between (middle) you have intermediaries , they
match Risk PMs on the asset side of their balance sheet
and Cash PMs on the liability side.
This is accountancy, each time a transaction of this kindis made, it has to be marked- to-market, thus all this ispegged to traded prices.
Source: Pozsar on Shadow Banking (2013)

You need such mechanisms to spread the risks across the system: with chance the net exposure of one
intermediary will meet the opposite hosted by another intermediary. You will net risk inside the system(Clearing is crucial).
CA Lehalle (Cours Bachelier, 2016)11 / 63

Intermediation of Risks

Going back to concepts mathematical finance is more familiar with:I

you are an investment bank, you sell a structured product or a derivative to clients;I

you do not hedge each book separately (or at least you shouldn’t): you hope to have other clientsconsuming other products flattening your (risk) inventory.

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Of course you will not succeed in netting 100% of the risk, hence you have to hedge the remaining book,in the markets (we hope they use optimal trading algorithms –i.e. continuous trading– to do this).

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But one step further: if you succeed into hedging continuously on markets (without liquidity, i.e. market
impact, issues), it just mean someone has the opposite risk in the market and hedges it on its side: you
should / could find it and net both positions (think about the crucial role of CCP here).

I

In this sense wrong way risk is not good for the liquidity on markets at all, you cannot believe you really
hedge if you impact the price.

Two good (but stylized) examples in the literature are [Stoikov and Saglam, 2009] and[Carmona and Webster, 2012].

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Sometimes Hedging is Far from Optimal

Even on liquid stocks and for vanilla options (close to maturity
in this case), hedging can go wrong.
The 19th of July 2012, a trading algorithms bought and soldshares every 30 minutes without any views on its market
impact [Lehalle et al., 2012].
For one visible mistake like this on liquid underlyings of vanilla
products, how many bad sophisticated hedging processes onless liquid (even OTC) markets...
Anonymous continuous hedging of a remaining position
outside of the bank does not mean all is going well.
Nevertheless we have solutions in recent literature:[Guéant and Pu, 2013], [Li and Almgren, 2014].
But nothing more generic, for instance the whole process ofhedging books in presence of wrong way risk is not studied(as far as I know). One step in this direction is[Schied and Zhang, 2013].
CA Lehalle (Cours Bachelier, 2016)13 / 63

A Modern Organization For an Intermediary

My advices to an investment bank:I

Net all your books , maintain two opposite positions is costly and risky,I

If you can’t it may be because you do not communicate enough internally (sometimes because of Chinese
walls...), hence be ready to hedge on the market ,I

But before try to match your small metaorders : send them to an internal place and cross them as muchas possible;

I

You will have synchronization issues (at the level of these metaorders, no reason to be synchronized), ask toyour traders to implement facilitation-like market making schemes inside the bank.

I

The remaining quantity has to be sent to markets as smoothly as possible, but it does not mean you will
have no impact. Who is your counterpart in the market should be an obsession: if you trade a one way
risk, you will pay for this in the future...

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The Crucial Role of Intermediaries

The are in charge ofI

anonymize the orders,I

find counterparts,

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provide unconflicted advices on the value of tradable instruments.

Investment banks have
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Flow desks,

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Market making desks,

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Execution and market access services,I

Financial analysis services.

Replication and hedging is a way to minimize the risk of intermediaries’ books. Ideally they should prefer to finda matching counterpart. Their efficiency deeply relies on their capability to net their positions to zero (i.e. to findbuyers once the sold, and the reverse). Most often residual risk will remain in their book, and they will have tohedge it anyway.

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Informational Role of intermediaries

As a regulator, you need intermediaries to:I

net and secure the positions,I

ensure a fair access to (fundamental) information,I

facilitate liquidity and make the markets.

About Financial Information:I

40 years ago, central banks’ decisions were not easy to accessI

20 years ago, you needed to by physically present during CEO/CFO speeches to have accurate information
about firms,

I

10 years ago, sector-wide information was not easy to accessI

today, cross-asset information is difficult to access.

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Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

CA Lehalle (Cours Bachelier, 2016)17 / 63

Buying and Selling in on the Market

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On an exchange or over the counter, if buyers and seller aredesynchronized, the price will not be efficient.

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Market Makers will “make the market” in selling to buyers at
Pt + ψ and buying to sellers at Pt − ψIthey earn the bid-ask spread ψ and take an inventory risk.

Market Makers Problematic
They earn the bid-ask spread and take an adverse selection risk:
what if the price is really changing? They will never buy-back at agood price...

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Intermediation of Liquidity

Intermediation is about providing unconflicted information, anonymity, price dissemination and concentrate thetrading flow (see [Scholtens and van Wensveen, 2008] and [Merton, 1995]).

Remark 1 (The Process of Intermediation of Liquidity)
If buyers and sellers of an instrument are not synchronized, intermediaries will answer to the first, take the risk intheir book, and wait for the other.

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Intermediation of Liquidity

Intermediation is about providing unconflicted information, anonymity, price dissemination and concentrate thetrading flow (see [Scholtens and van Wensveen, 2008] and [Merton, 1995]).

Remark 1 (The Process of Intermediation of Liquidity)
If buyers and sellers of an instrument are not synchronized, intermediaries will answer to the first, take the risk intheir book, and wait for the other.
If the price does not move that much and buyer and sellers come according to a Poisson process, things should
be ok. Nevertheless intermediaries’ books face a risk (what if the price move between two arrivals?).
New Matching Technologies? If you want to go further: you can think about a world in which all thematchings were automated, taking places on electronic venues (i.e. trading facilities). Even sophisticatedproducts could be matched electronically.
Just have a look at ripple and blockchain (the protocols underlying the Bitcoin crypto-currency), or
ethereum.org (another protocol dedicated to complex contracts).

CA Lehalle (Cours Bachelier, 2016)18 / 63

Market Making 101
When a market maker says “yes” to every sollicitation...

Uncontrolled Inventory of a Market Maker (MM)
If the arrival of buyers (resp. sellers) follows a Poisson process with intensity λ− (resp. λ− ), when a MM iscontinuously present at the best bid and ask, its inventory I asymptotically follows the Brownian motionp(1)
dI = (λ+ − λ− )dt + λ+ + λ− · dWt .
It is enough to note the dynamics of inventory I are dI = dN + − dN − and to refer to the construction ofBrownian motion using a Poisson process. Hence a MM has to ask herself if λ+ = λ− ? i.e. is my client flow unbalanced ? do they have (private)
information?
Moreover, she even cannot afford to have a diffusive inventory (i.e. the dW part), hence she will control its
inventory using a 2D parameter (δ + , δ − ) to obtained as dynamics:
dI = dN + (δ + ) − dN − (δ − ).
→ what could be (δ + , δ − ) ?
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Controlling the Risk

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Regulators want the market makers to control the risk they accumulate in the system.

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But they need to allow them enough inventory to wait for the next participant,I

Otherwise the market will freeze

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Market Making: First Attempt to Control the Risk

The Market Maker Controls
The MM wants to decrease λ+ and increase λ− (resp. λ− /λ+ ) when her inventory is positive (resp. negative).
When it is possible to use (δ + , δ − ) so that λ+ (δ + ) − λ− (δ − ) ∝ −I, then the market maker inventory dynamics
are Ornstein-Uhlenbeck:dI = −γI dt + σdW ,and hence asymptotically I ∼ N (0, σ 2 /(2γ)).
This a way to control the risk of a market making strategy. Several papers have been written on thisI

Optimal dealer pricing under transactions and return uncertainty – [Ho and Stoll, 1981] (economics)

I

High-frequency trading in a limit order book – [Avellaneda and Stoikov, 2008] (first quantitative model)

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Dealing with the inventory risk: a solution to the market making problem – [Guéant et al., 2013] (full solution)

I

Market Impacts and the Life Cycle of Investors Orders – [Bacry et al., 2015] (toy model using Hawkes
processes)I

see Olivier Guéant’s book [Guéant, 2016] for details.

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Regulation vs. Liquidity

Regulators do not want to let intermediary take too much risk because it is very difficult to check what they
have in it.
Assume this toy model: buy and sell orders intensity are the same λ− = λ+ =: λ, then the inventory of a market
maker (MM) accepting all trades evolves like√dI = 2λ dW .
| {z }σIf a regulator set a “risk limit” at Aq σ (say it is the qth quantile of a Gaussian: Aq := Φ(q)), then as soon as|I| > Aq σ, the MM stops to accept trades in one of the two directions, she has to wait on average λ units of time.

Freezing Market Makers
In our toywhen model: the regulator set the risk limit at R := Aq√
as Φ−1 R/( 2λ) > λ.

CA Lehalle (Cours Bachelier, 2016)√

2λ the market “freezes on average” as soon

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Adverse Selection

As we just have seenI

Adverse selection is the main issue a market maker should
face.

I

On the other side, the liquidity taker will face opportunity cost:you buy now, but few minutes later, the price would have been
better.

If the market maker only deals with someone having directional
information for 100% of the trades, the market maker will bankrupt.
Market makers do not usually invest in extracting “fundamental” (i.e.
exogenous) information. Nevertheless

CA Lehalle (Cours Bachelier, 2016)

I

Investment banks have analysts and multiple clients,

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Systematic market makers have newsfeeds.

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Making the Market in Front of Adverse Selection: Kyle’s 85 Model

The framework

CA Lehalle (Cours Bachelier, 2016)

I

An informed trader, knowing the future priceI

Noise traders, knowing nothingI

A market makers, having only access to distributions (thanks to“backtests” / observations); she changes her price linearly according to
the price pressure she observes: fP (q) = P̃ + λ · q.

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Making the Market in Front of Adverse Selection: Kyle’s 85 Model

The frameworkI

An informed trader, knowing the future priceI

Noise traders, knowing nothingI

A market makers, having only access to distributions (thanks to“backtests” / observations); she changes her price linearly according to
the price pressure she observes: fP (q) = P̃ + λ · q.

I

The informed trader adjusts his participation to maximize its profit (given P̃
and λ),The informed trader chooses his
quantity to maximize his expected
profitCA Lehalle (Cours Bachelier, 2016)24 / 63

Making the Market in Front of Adverse Selection: Kyle’s 85 Model

The framework

The market maker choose P̃ andλ to adjust her price to the flow

CA Lehalle (Cours Bachelier, 2016)

I

An informed trader, knowing the future priceI

Noise traders, knowing nothingI

A market makers, having only access to distributions (thanks to“backtests” / observations); she changes her price linearly according to
the price pressure she observes: fP (q) = P̃ + λ · q.

I

The informed trader adjusts his participation to maximize its profit (given P̃
and λ),I

The market makers know the distribution of the informed price and set P̃
and λ so that her price is as close as possible to its expectation.

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Making the Market in Front of Adverse Selection: Kyle’s 85 Model

Following Continuous Auctions and Insider Trading – [Kyle, 1985]:
I

Remember the market makers fear adverse selection.

I

We have informed traders, they know the price will be pω after their trade, pω ∼ N (P ∗ , σp2 ).

I

Other traders, (i.e. noise traders for Kyle) trade for other reasons, their net direction is nω ∼ N (0, σn2 ).

I

The informed traders have to choose a participation Q(p) (they know p) to maximize their profit,

I

Knowing the market makers (MM) will react to the net perceived flow linearly: the public price will befP (Q(pω ) + nω ) = P̃ + λ · (Q(pω ) + nω ).

I

Moreover in their filtration, the MM should produce a price being the best estimator of pω given
Q(pω ) + nω .

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒
I

They decide to trade Q(pω ) = (pω − P̃)/(2λ).
The MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :CA Lehalle (Cours Bachelier, 2016)

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :CA Lehalle (Cours Bachelier, 2016)


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )26 / 63

The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :CA Lehalle (Cours Bachelier, 2016)


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )

⇒



 P̃

=


 λ

=

P∗
σp2 /(2λ)σp2 /(2λ)2 + σn2

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :

⇒


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )

⇒



 P̃

=


 λ

=

P∗
σp2 /(2λ)σp2 /(2λ)2 + σn2

It can be solved with λ = σp /(2σn ).

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The Economics of Market Impact: Kyle’s 85 Solution

I

Informed traders maximize their expected price: arg max E((pω − fp (Q + nω ))Q|pω ).
Q⇒

They decide to trade Q(pω ) = (pω − P̃)/(2λ).

IThe MMs have to choose P̃ and λ so that P̃ + λ · (Q(pω ) + nω ) = E (pω |Q(pω ) + nω ).

I

The solution is the linear regression of p on Q(p) + nω :

⇒


 P∗

=

 λ

=

P̃ + λE(Q(pω ) + nω )Cov(p, Q(p) + nω )V(Q(p) + nω )

⇒



 P̃

=


 λ

=

P∗
σp2 /(2λ)σp2 /(2λ)2 + σn2

It can be solved with λ = σp /(2σn ).

I

The more potential informational price move (i.e. large σp ), the largest impact.

I

The more non informative flow, the more difficult for the MM to identify information, hence the less she
impact the price.

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It is a Little More Complex in PracticeOn our database of 300,000 large orders
Market Impact takes place in different phasesI

the transient impact, concave in time,I

reaches its maximum, the temporary impact, at the end of themetaorder,I

then it decays,I

up to a stationary level; the price moved by a permanent shift.

In [Bacry et al., 2015] we studied all the phases, using intraday and daily analysis (for the first time). We
underlined the importance of some “normalization variables”: the uncertainty on the price formation process , thecapability of the orderbook to resist to volume pressure , and the duration of the metaorder.
Following [Waelbroeck and Gomes, 2013] and simultaneously with [Brokmann et al., 2014], we proposed an
explanation of permanent impact .

CA Lehalle (Cours Bachelier, 2016)27 / 63

The Price Formation Process

Ticker Machine

Ticker Tape

The price is not only the outcomeof the balance between liquidity
providers and liquidity consumers(not really between buyers andsellers from our perspective). It is
an input for most agents decision
processes. They all pay a lot of
attention to the price (most of thepositions have to be marked tomarket). The price of atransaction has more importancethan any indicative one.

In theory market makers provide liquidity and fear adverse selection. Their protection is to impact the price (i.e.
to statistically guess how much information is in the flow).
We can select very specific dates with exogenous information or flow and look at the prices and volumes.

CA Lehalle (Cours Bachelier, 2016)28 / 63

That is real stuff

This is the effect of the unexpected announcement by president Sarkozy (France, 2008) |that public TV channels
will no more be allowed to sell advertising... The value of the main french private Channels jumped immediately.
CA Lehalle (Cours Bachelier, 2016)29 / 63

Propagation - correlations

Bouygues price around the announcement of the end of advertising on public channels (8 Jan.
2008).
CA Lehalle (Cours Bachelier, 2016)30 / 63

Propagation - zooming out: 30 minutes

CA Lehalle (Cours Bachelier, 2016)31 / 63

Propagation - zooming out: 1 hour

CA Lehalle (Cours Bachelier, 2016)32 / 63

Propagation - zooming out: daily

CA Lehalle (Cours Bachelier, 2016)33 / 63

Propagation - sector: daily media

CA Lehalle (Cours Bachelier, 2016)34 / 63Effets of Such an Unexpected Information

I

The price changed almost immediately (up), with large volumes, but oscillated a little bit

I

Other (related, correlated) stocks, moved accordingly

I

Sectorial moves drove the prices down.

We saw a jump that can be explained at 100% by an exogenous (i.e. “fundamental”) information. We could say
information drove prices . In such a case there is no proof of causality like: buying pressure → price move. It
is more a common agreement on a new price.

CA Lehalle (Cours Bachelier, 2016)35 / 63

But When the Information is Hidden?
In [Besson and Lehalle, 2014], we try to split price moves caused by trades or just “quote shifts”, we provided
aggregated statistics but here are two anecdotical cases, the 18th of Oct. 2012; it shows how news impact theprice, it somehow shows participants have several ways to impulse permanent impact:easy to process news

CA Lehalle (Cours Bachelier, 2016)

difficult to process news

36 / 63

And Now an Exogenous Flow with No Information

Market impact generated by the unwind of Jerome Kerviel’s position
CA Lehalle (Cours Bachelier, 2016)37 / 63

We Have Seen Few Different Cases

TF1 — “Pure” information : the causality is information → prices → volumes → oscillations,
Ericsson — Information, easy to process : prices did not needed transactions to move,
Nokia — Information, but difficult to process : prices moved because of transactions pressure,
Kerviel — No information (i.e. “Cash Trades”) the causality is for sure volumes → prices.
Where does the “permanent impact” comes from? Is it
I

Price discovery ? price “had to goe there”,

I

or Price Formation ? mechanical pressure of the traded volume on the prices.

CA Lehalle (Cours Bachelier, 2016)38 / 63

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

CA Lehalle (Cours Bachelier, 2016)

39 / 63

The Market Impact of Large Orders

in [Moro et al., 2009]Market Impact takes place in different phases

CA Lehalle (Cours Bachelier, 2016)I

the transient impact, concave in time,I

reaches its maximum, the temporary impact, at the end of themetaorder,I

then it decays,I

up to a stationary level; the price moved by a permanent shift.

39 / 63

The Market Impact of Large Orders

On our database of 300,000 large orders[Bacry et al., 2015]
Market Impact takes place in different phasesI

the transient impact, concave in time,I

reaches its maximum, the temporary impact, at the end of themetaorder,I

then it decays,I

up to a stationary level; the price moved by a permanent shift.

f

CA Lehalle (Cours Bachelier, 2016)

39 / 63

The Market Impact of Large Orders

To be more than anecdotical, it is needed to make statistics, that for we need a not of occurrences of“ metaorders ”. Some paper documented the “square root impact”: the temporary impact of your flow is
proportional to the its square root. But the three phases has been studied in fewer papers:
I

The Non-Linear Market Impact of Large Trades: Evidence from Buy-Side Order Flow
[Bershova and Rakhlin, 2013] – intraday impact

I

Is Market Impact a Measure of the Information Value of Trades? Market Response to Liquidity vs. Informed
Trades [Waelbroeck and Gomes, 2013] – daily impact of cash trades

I

Slow decay of impact in equity markets [Brokmann et al., 2014] – daily impact of informed trades (hedge
fund)

I

Market Impacts and the Life Cycle of Investors Orders [Bacry et al., 2015] – intraday and daily impact of
informed trades (bank)

CA Lehalle (Cours Bachelier, 2016)

40 / 63

The Square Root Effect

3: Trace of the duration Y = T on the residuals of the X = R daily participation regression. Top: using
metric, bottom: using a L1 metric.
Regression(R.1)

Parameter Coef. (log-log) Coef. (L2) Coef. (L1)
Daily participation

0.54

0.45

0.40

Daily participation
Duration

0.59
0.23

0.54
0.35

0.59
0.23

Daily participation
Bid-ask spread

0.44
0.28

Daily participation
Volatility

0.53
0.96

Trading rate

0.43

0.33

0.43

I

Trading rate
Duration

0.37
0.15

0.56
0.24

0.45
0.23

did you trading process reacted to market
conditions?

I

are you alone?

I

etc.

(R.2)

(R.3)

(R.4)

(R0 .1)(R0 .2)(R0 .3)(R0 .4)

Trading rate
Bid-ask spread

0.32
0.57

Trading rate
Volatility

0.32
0.88

The Formula should be close tos
Traded volume
MI ∝ σ ·· T −0.2
Daily volume
The term in duration is very difficult to estimate because
you have a lot of conditioning everywhere:

We used different methods.

Source:al., 2015]
2: Direct regression approachalgorithm [Bacry
describedet
in Section
2.4 of the temporary market impact fors setsCAof Lehalle
explanatory
For each set, the power exponent estimations are given using L1 dis(Coursvariables.
Bachelier, 2016)41 / 63

Permanent Market Impact

We had enough data to investigate long term impact, exploring the relationships between permanent impact andtraded information.

CA Lehalle (Cours Bachelier, 2016)42 / 63

Permanent Market Impact

We had enough data to investigate long term impact, exploring the relationships between permanent impact andtraded information.
Daily price moves

CA Lehalle (Cours Bachelier, 2016)

I

If you plot the long term price moves (x-axis indays), you see an regular increase;I

But the same stock is traded today, tomorrow, theday after, etc.

42 / 63

Permanent Market Impact

We had enough data to investigate long term impact, exploring the relationships between permanent impact andtraded information.
Daily price moves
I If you plot the long term price moves (x-axis indays), you see an regular increase;

CA Lehalle (Cours Bachelier, 2016)

I

But the same stock is traded today, tomorrow, theday after, etc.

I

Once you remove the market impact of “future”
trades (similarly to[Waelbroeck and Gomes, 2013]), you obtain adifferent shape.

I

If you look each curve: the yellow one contains the
CAPM β (the metaorders are trading market-wide
moves), the green curve contains the idiosyncratic
moves, this shape can be read as the daily decayof metaorders impact.

42 / 63

Informational vs. Mechanical Moves of the Price

Remark 2
The main source of persistence of price move after the market impact (i.e. trades of large orders) is
informational:I“factors” for slow, beta-driven investors,

Ialpha for hedge funds.

Keep in mind the difference between TF1 jump (informational) and Kerviel’s inventory unwinding (mechanical).

CA Lehalle (Cours Bachelier, 2016)43 / 63

Transient Market Impact

We only have conjectures, but we know (at left) liquidity
providers are not “telepathic” (same impact for the first
MM minutes regardless to the duration of themetaorder). Nevertheless the concavity
I

could come from “front running”,

I

could come from “auto selection of market makers”,

I

can be explained by an Hawkes process.

See [Bacry et al., 2015] for details.

CA Lehalle (Cours Bachelier, 2016)44 / 63

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

CA Lehalle (Cours Bachelier, 2016)45 / 63

Price Impact and the Propagator ModelAt the highest frequency, it is possible to define the Price Impact (the pricemove just after a trade, cf. [Wyart et al., 2008] and [Gatheral, 2010]) as:∆P̃0 (t) := E(0 (P(t) − P(0))|trade at 0).

The Propagator Model∆P̃0 (t) = η(Q0 )K (t − 0) where K (·) is a Kernel, usually taken asIan exponential kernel: K (s) = λ exp −(s/λ);I

a power law: K (s) = (1 + s)−γ /(γ − 1).

Autoregressive models can be use as propagators.

CA Lehalle (Cours Bachelier, 2016)45 / 63

Price Impact and the Propagator ModelAt the highest frequency, it is possible to define the Price Impact (the pricemove just after a trade, cf. [Wyart et al., 2008] and [Gatheral, 2010]) as:∆P̃0 (t) := E(0 (P(t) − P(0))|trade at 0).

The Propagator Model∆P̃0 (t) = η(Q0 )K (t − 0) where K (·) is a Kernel, usually taken asIan exponential kernel: K (s) = λ exp −(s/λ);I

a power law: K (s) = (1 + s)−γ /(γ − 1).

Autoregressive models can be use as propagators.

CA Lehalle (Cours Bachelier, 2016)45 / 63

Price Impact and the Propagator ModelAt the highest frequency, it is possible to define the Price Impact (the pricemove just after a trade, cf. [Wyart et al., 2008] and [Gatheral, 2010]) as:∆P̃0 (t) := E(0 (P(t) − P(0))|trade at 0).

The Propagator Model∆P̃0 (t) = η(Q0 )K (t − 0) where K (·) is a Kernel, usually taken asIan exponential kernel: K (s) = λ exp −(s/λ);I

a power law: K (s) = (1 + s)−γ /(γ − 1).

Autoregressive models can be use as propagators.

CA Lehalle (Cours Bachelier, 2016)45 / 63

Price Impact and the Propagator ModelAt the highest frequency, it is possible to define the Price Impact (the pricemove just after a trade, cf. [Wyart et al., 2008] and [Gatheral, 2010]) as:∆P̃0 (t) := E(0 (P(t) − P(0))|trade at 0).

The Propagator Model∆P̃0 (t) = η(Q0 )K (t − 0) where K (·) is a Kernel, usually taken asIan exponential kernel: K (s) = λ exp −(s/λ);I

a power law: K (s) = (1 + s)−γ /(γ − 1).

Autoregressive models can be use as propagators.

CA Lehalle (Cours Bachelier, 2016)45 / 63

Price Impact and the Propagator ModelAt the highest frequency, it is possible to define the Price Impact (the pricemove just after a trade, cf. [Wyart et al., 2008] and [Gatheral, 2010]) as:∆P̃0 (t) := E(0 (P(t) − P(0))|trade at 0).

The Propagator Model∆P̃0 (t) = η(Q0 )K (t − 0) where K (·) is a Kernel, usually taken asIan exponential kernel: K (s) = λ exp −(s/λ);I

a power law: K (s) = (1 + s)−γ /(γ − 1).

Autoregressive models can be use as propagators.

CA Lehalle (Cours Bachelier, 2016)45 / 63

Price Impact vs. Adverse Selection

Adverse selection is the fact you obtained liquidity now and you could
obtain it later and have a better price.

CA Lehalle (Cours Bachelier, 2016)46 / 63

Price Impact vs. Adverse Selection

Adverse selection is the fact you obtained liquidity now and you could
obtain it later and have a better price.

Price Impact and Adverse Selection
Price Impact for market orders implies Adverse selection for limit orders.
If the price in martingale after a price change, there is adverse selection ; theimbalance says you have a little less adverse selection than that since once afull tick has been consumed, they are chances the discovered quantity islarger than average.
When you owns a limit order in the book: the more orders behind you, themore protected vs. adverse selection.
2nd worst kept secret of HFT : if you have too few orders behind you, cancel your limit order.
“discovered quantity” means the quantity at second limit that is now a first limit

CA Lehalle (Cours Bachelier, 2016)46 / 63Effect of aggressive orders I

When an order is sent aggressively to the market, as far as it is higher than the quantity available at the first limit
in the market:I

the price changes immediately (most studies have shown that is follows a power law [Wyart et al., 2008]),I

the market provides decay.

This is clearly stated into Jim Gatheral’s paper [Gatheral, 2010] this way: given that an order of size v hit
aggressively the market at time τ , its remaining impact on the market at any t greater than τ is:(2)

δt (v , τ ) = ητ (v ) G(t − τ )where ηt is the immediate market impact function and G a decay kernel.
Consequently, the cumulated effect of all trades (τ, vτ ) that occured before t is:Z∆t (v , τ ) =ητ (vτ ) G(t − τ ) dη(τ, vτ )
τ <t

The decay comes from liquidity providers that agree to insert orders between the actual price impacted by theorder of size δv and the previous price.

CA Lehalle (Cours Bachelier, 2016)

47 / 63Effect of aggressive orders II

This can be modelled using a fair value for the price Stf . The decay is then a functional of difference between themarket price St and this fair price:δt (v , τ ) = ητ (vτ ) G(t − τ, |St − Stf |)given that an order of size δvτ has been sent aggressively to the market at τ for a lot of τ in the past, and if thefair price is modelled by a stochastic process Stf , we have:Z(3)

St = S0 +τ <t

CA Lehalle (Cours Bachelier, 2016)ητ (vτ ) G(t − τ, |St − Stf |) dη(τ, vτ )48 / 63Effect of passive ordersI

On average and market wide, there is no reason market orders and limit orders have a different impact on
the priceI

If you consider there is one limit order for one market order,I

We will see (orderbook dynamics) that is a little more complex.

I

You have to consider the whole trajectories of orders to understand the real cost of a market or a limit order.

I

At this stage we can just look at static pictures, conditioned by a given stopping time and the owner of theorder.

CA Lehalle (Cours Bachelier, 2016)49 / 63

Price Profiles
Price profiles: future and past of the mid-price (solid) or
bid and asks (dotted), conditionally to an execution.
Global Banks
Instit. Brokers
HF MM
HF Prop.

0.04

Average mid-price move

0.02

0.00

I

Instutional Brokers (i.e. essentially “client flows”:
with a decision taken at a daily scale and large
metaorders)I

HFT, split in HF market makers and HF proprietary
tradersI

Global banks, having a mix of client flows andproprietary trading flows.

I

Conditionally to the owner of the order , theprofile can be very different

I

You can compare such graphs and make matrices[Brogaard et al., 2012]I

Nevertheless the big picture is dynamic...

−0.02
−0.04
−0.06
−0.08
−0.10
−100

0

CA Lehalle (Cours Bachelier, 2016)

100
200
Number of trades

300

400

50 / 63

Imbalance Profiles

0.1

0.0

Average Imbalance−0.1

Imbalance profiles: State of the bookconditionally to an execution, renormalized
such as best opposite is 1, the green bar isyour order size.

−0.2
−0.3
−0.4
−0.5
−0.6

Instit. Brokers

CA Lehalle (Cours Bachelier, 2016)

Global Banks

HF MM

HF Prop.

51 / 63

From High Frequency to Low Frequencies

If someone trade at a given frequency 1/δt from 0, his price impact at K δt willbe (for an exponential kernel)
X
P(K δt) − P(0) =η(1)λe−k δt λ ' η(1)(1 − e−K δt λ )/δt.
k ≤KAnd for a power lawP(K δt) − P(0) = η(1) 1 − (1 + K δt)−(γ−1) /δt.
In both cases, if he stops trading at K δt, the price will fully revert according to
an exponential (or a power lax).

Transient Impact and Decay
The concave increase of the impact with time and its reversion can be
explained using propagator models.

CA Lehalle (Cours Bachelier, 2016)52 / 63

From High Frequency to Low Frequencies

If someone trade at a given frequency 1/δt from 0, his price impact at K δt willbe (for an exponential kernel)
X
P(K δt) − P(0) =η(1)λe−k δt λ ' η(1)(1 − e−K δt λ )/δt.
k ≤KAnd for a power lawP(K δt) − P(0) = η(1) 1 − (1 + K δt)−(γ−1) /δt.
In both cases, if he stops trading at K δt, the price will fully revert according to
an exponential (or a power lax).

Transient Impact and Decay
The concave increase of the impact with time and its reversion can be
explained using propagator models.

CA Lehalle (Cours Bachelier, 2016)52 / 63

From High Frequency to Low Frequencies

If someone trade at a given frequency 1/δt from 0, his price impact at K δt willbe (for an exponential kernel)
X
P(K δt) − P(0) =η(1)λe−k δt λ ' η(1)(1 − e−K δt λ )/δt.
k ≤KAnd for a power lawP(K δt) − P(0) = η(1) 1 − (1 + K δt)−(γ−1) /δt.
In both cases, if he stops trading at K δt, the price will fully revert according to
an exponential (or a power lax).

Transient Impact and Decay
The concave increase of the impact with time and its reversion can be
explained using propagator models.

CA Lehalle (Cours Bachelier, 2016)52 / 63

Common sense for Temporary Market Impact modelI

Crossing the Bid-Ask spread. If the last trade was on the opposite side to my order (once over two trades...),
I pay the spread: 2ψ.

CA Lehalle (Cours Bachelier, 2016)

53 / 63

Common sense for Temporary Market Impact modelI

Crossing the Bid-Ask spread. If the last trade was on the opposite side to my order (once over two trades...),
I pay the spread: 2ψ.

I

Dynamical market depth. The more the volatility of the price diffusion is high, the more the LOB is
structurally empty: M.I. increases with σ.

CA Lehalle (Cours Bachelier, 2016)

53 / 63

Common sense for Temporary Market Impact modelI

Crossing the Bid-Ask spread. If the last trade was on the opposite side to my order (once over two trades...),
I pay the spread: 2ψ.

I

Dynamical market depth. The more the volatility of the price diffusion is high, the more the LOB is
structurally empty: M.I. increases with σ.

I

Usual quantities: the more my volume v is large compared to the “usual” traded volume V∆T (during thistime interval ∆T ), the more my M.I. increases.

CA Lehalle (Cours Bachelier, 2016)

53 / 63

Market Impact main modelWith ψ the half spread, σ the volatility (in currency), and V a constant homogeneous with a quantity of shares,
The most common model is:
 γ
v
(4)
ηt (v ) = αψt + κσt ·Vt
Usually you take γ = 1/2, κ is a function of your dataset (it corrects the denominator) and α is a function of yourtrading signals.

CA Lehalle (Cours Bachelier, 2016)

54 / 63

Implicit Orderbook−+
Let fLOB(s) be the volume available in the LOB by sellers (ask prices) at price s (resp. fLOB(s) for bids quantities).
When I want to buy a quantity V of shares of this instrument, I will generate a Market Impact of SMI implicitly
defined by the equation (S0 is the last quoted price):

Z SMI
(5)

V =
s=S0

−
fLOB
(s) ds.

SMI of (5) is the market Impact η(V ) associated to the volume V .

[Property: Implicit Orderbook] Formal derivation of the market impact equation (5) with respect to V gives the
Implicit Orderbook associated to the Market Impact model η:
(6)

−
fLOB
(SPMI (v )) =1∂v η(v )Tell me your price impact function SMI (v ), I will tell you your implicit orderbook function. So if you think thatmarket impact is constant, it means that you think that LOB is infinite.
CA Lehalle (Cours Bachelier, 2016)

55 / 63

Simple Checks For The Power Law Models
For the generic power law market impact, we obtain:(7)V

fc (S0 + h) =

κσγ

h−ψ
κσ

 γ−1
γ

(at left: Market impact and order book functions for several values of γ)
Equation (7) implies that γ ≤ 1, besides:

A square root market impact
A square root market impact implies a linear implicit LOB.

With γ = 1/2, we obtain
fc (S0 + h) = 2(h − ψ)VCA Lehalle (Cours Bachelier, 2016)56 / 63

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

CA Lehalle (Cours Bachelier, 2016)57 / 63

Stylized Facts on Liquidity

CA Lehalle (Cours Bachelier, 2016)57 / 63

Outline

1

The Financial System as a Network of Intermediaries
Risks Transformation as The Primary Role of The Financial System
Making the Market: the Stakes of Liquidity Provision
The Market Impact of Large Orders
Quant Models For Common Practices

2

Stylized Facts on Liquidity

3

Optimal Trading

CA Lehalle (Cours Bachelier, 2016)

58 / 63

Optimal Trading

CA Lehalle (Cours Bachelier, 2016)

58 / 63

Thank You For Your Attention

CA Lehalle (Cours Bachelier, 2016)59 / 63References I
Avellaneda, M. and Stoikov, S. (2008).
High-frequency trading in a limit order book.
Quantitative Finance, 8(3):217–224.
Bacry, E., Iuga, A., Lasnier, M., and Lehalle, C.-A. (2015).
Market Impacts and the Life Cycle of Investors Orders.
Market Microstructure and Liquidity, 1(2).
Bershova, N. and Rakhlin, D. (2013).
The Non-Linear Market Impact of Large Trades: Evidence from Buy-Side Order Flow.
Social Science Research Network Working Paper Series.
Besson, P. and Lehalle, C.-A. (2014).
The Deal/Book Split Analysis: A New Method to Disentangle the Contribution to Market and Limit Orders in Any Price Change.
Social Science Research Network Working Paper Series.
Brogaard, J., Hendershott, T. J., and Riordan, R. (2012).
High Frequency Trading and Price Discovery.
Technical report, Berkeley University.
Brokmann, X., Serie, E., Kockelkoren, J., and Bouchaud, J. P. (2014).
Slow decay of impact in equity markets.
Cardaliaguet, P. and Lehalle, C.-A. (2016).
Mean Field Game of Controls and An Application To Trade Crowding.
Carmona, R. and Webster, K. (2012).
High Frequency Market Making.

CA Lehalle (Cours Bachelier, 2016)60 / 63References II

Cartea, A., Jaimungal, S., and Penalva, J. (2015).
Algorithmic and High-Frequency Trading (Mathematics, Finance and Risk).
Cambridge University Press, 1 edition.
Gatheral, J. (2010).
No-Dynamic-Arbitrage and Market Impact.
Quantitative Finance, 10(7).
Guéant, O. (2016).
The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making.
Chapman and Hall/CRC.
Guéant, O., Lehalle, C.-A., and Fernandez-Tapia, J. (2013).
Dealing with the inventory risk: a solution to the market making problem.
Mathematics and Financial Economics, 4(7):477–507.
Guéant, O. and Pu, J. (2013).
Option pricing and hedging with execution costs and market impact.
Ho, T. and Stoll, H. R. (1981).
Optimal dealer pricing under transactions and return uncertainty.
Journal of Financial Economics, 9(1):47–73.
Kyle, A. P. (1985).
Continuous Auctions and Insider Trading.
Econometrica, 53(6):1315–1335.

CA Lehalle (Cours Bachelier, 2016)

61 / 63References III
Lachapelle, A., Lasry, J.-M., Lehalle, C.-A., and Lions, P.-L. (2016).
Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis.
Mathematics and Financial Economics, 10(3):223–262.
Lehalle, C.-A., Laruelle, S., Burgot, R., Pelin, S., and Lasnier, M. (2013).
Market Microstructure in Practice.
World Scientific publishing.
Lehalle, C.-A., Lasnier, M., Bessson, P., Harti, H., Huang, W., Joseph, N., and Massoulard, L. (2012).
What does the saw-tooth pattern on US markets on 19 July 2012 tell us about the price formation process.
Technical report, Crédit Agricole Cheuvreux Quant Note.
Li, T. M. and Almgren, R. (2014).
Option Hedging with Smooth Market Impact.
Technical report.
Merton, R. C. (1995).
A Functional Perspective of Financial Intermediation.
Financial Management, 24(2):23+.
Moro, E., Vicente, J., Moyano, L. G., Gerig, A., Farmer, D. J., Vaglica, G., Lillo, F., and Mantegna, R. N. (2009).
Market impact and trading profile of large trading orders in stock markets.
Schied, A. and Zhang, T. (2013).
A state-constrained differential game arising in optimal portfolio liquidation.
Scholtens, B. and van Wensveen, D. (2008).
The theory of financial intermediation: an essay on what it does (not) explain.
Elsevier Science, 1 edition.

CA Lehalle (Cours Bachelier, 2016)62 / 63References IV

Stoikov, S. and Saglam, M. (2009).
Option market making under inventory risk.
Waelbroeck, H. and Gomes, C. (2013).
Is Market Impact a Measure of the Information Value of Trades? Market Response to Liquidity vs. Informed Trades.
Social Science Research Network Working Paper Series.
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Relation between Bid-Ask Spread, Impact and Volatility in Double Auction Markets.
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